Correlated Equilibria and Communication in Games Françoise Forges
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Lecture Notes
GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies .................................. -
Kranton Duke University
The Devil is in the Details – Implications of Samuel Bowles’ The Moral Economy for economics and policy research October 13 2017 Rachel Kranton Duke University The Moral Economy by Samuel Bowles should be required reading by all graduate students in economics. Indeed, all economists should buy a copy and read it. The book is a stunning, critical discussion of the interplay between economic incentives and preferences. It challenges basic premises of economic theory and questions policy recommendations based on these theories. The book proposes the path forward: designing policy that combines incentives and moral appeals. And, therefore, like such as book should, The Moral Economy leaves us with much work to do. The Moral Economy concerns individual choices and economic policy, particularly microeconomic policies with goals to enhance the collective good. The book takes aim at laws, policies, and business practices that are based on the classic Homo economicus model of individual choice. The book first argues in great detail that policies that follow from the Homo economicus paradigm can backfire. While most economists would now recognize that people are not purely selfish and self-interested, The Moral Economy goes one step further. Incentives can amplify the selfishness of individuals. People might act in more self-interested ways in a system based on incentives and rewards than they would in the absence of such inducements. The Moral Economy warns economists to be especially wary of incentives because social norms, like norms of trust and honesty, are critical to economic activity. The danger is not only of incentives backfiring in a single instance; monetary incentives can generally erode ethical and moral codes and social motivations people can have towards each other. -
Collusion Constrained Equilibrium
Theoretical Economics 13 (2018), 307–340 1555-7561/20180307 Collusion constrained equilibrium Rohan Dutta Department of Economics, McGill University David K. Levine Department of Economics, European University Institute and Department of Economics, Washington University in Saint Louis Salvatore Modica Department of Economics, Università di Palermo We study collusion within groups in noncooperative games. The primitives are the preferences of the players, their assignment to nonoverlapping groups, and the goals of the groups. Our notion of collusion is that a group coordinates the play of its members among different incentive compatible plans to best achieve its goals. Unfortunately, equilibria that meet this requirement need not exist. We instead introduce the weaker notion of collusion constrained equilibrium. This al- lows groups to put positive probability on alternatives that are suboptimal for the group in certain razor’s edge cases where the set of incentive compatible plans changes discontinuously. These collusion constrained equilibria exist and are a subset of the correlated equilibria of the underlying game. We examine four per- turbations of the underlying game. In each case,we show that equilibria in which groups choose the best alternative exist and that limits of these equilibria lead to collusion constrained equilibria. We also show that for a sufficiently broad class of perturbations, every collusion constrained equilibrium arises as such a limit. We give an application to a voter participation game that shows how collusion constraints may be socially costly. Keywords. Collusion, organization, group. JEL classification. C72, D70. 1. Introduction As the literature on collective action (for example, Olson 1965) emphasizes, groups often behave collusively while the preferences of individual group members limit the possi- Rohan Dutta: [email protected] David K. -
Analyzing Just-In-Time Purchasing Strategy in Supply Chains Using an Evolutionary Game Approach
Bulletin of the JSME Vol.14, No.5, 2020 Journal of Advanced Mechanical Design, Systems, and Manufacturing Analyzing just-in-time purchasing strategy in supply chains using an evolutionary game approach Ziang LIU* and Tatsushi NISHI** *Graduate School of Engineering Science, Osaka University 1-3 Machikaneyama-Cho, Toyonaka City, Osaka 560-8531, Japan E-mail: [email protected] **Graduate School of Natural Science and Technology, Okayama University 3-1-1 Tsushima-naka, Kita-ku, Okayama City, Okayama 700-8530, Japan Received: 18 December 2019; Revised: 30 March 2020; Accepted: 9 April 2020 Abstract Many researchers have focused on the comparison between the JIT model and the EOQ model. However, few of them studied this problem from an evolutionary perspective. In this paper, a JIT purchasing with the single- setup-multi-delivery model is introduced to compare the total costs of the JIT model and the EOQ model. Also, we extend the classical JIT-EOQ models to a two-echelon supply chain which consists of one manufacturer and one supplier. Considering the bounded rationality of players and the quickly changing market, an evolutionary game model is proposed to discuss how these factors impact the strategy selection of the companies. And the evolutionarily stable strategy of the proposed model is analyzed. Based on the analysis, we derive the conditions when the supply chain system will choose the JIT strategy and propose a contract method to ensure that the system converges to the JIT strategy. Several numerical experiments are provided to observe the JIT and EOQ purchasing strategy selection of the manufacturer and the supplier. -
Kahneman's Thinking, Fast and Slow from the Standpoint of Old
Kahneman’s Thinking, Fast and Slow from the Standpoint of Old Behavioural Economics (For the 2012 HETSA Conference) Peter E. Earl School of Economics, University of Queensland, St Lucia, Brisbane, QLD 4072, Australia, email: [email protected] Abstract Daniel Kahneman’s bestseller Thinking, Fast and Slow presents an account of his life’s work on judgment and decision-making (much of it conducted with Amos Tversky) that has been instrumental in the rise of what Sent (2004) calls ‘new behavioural economics’. This paper examines the relationship between Kahneman’s work and some key contributions within the ‘old behavioural’ literature that Kahneman fails to discuss. It show how closely aligned he is to economic orthodoxy, examining his selective use of Herbert Simon’s work in relation to his ‘two systems’ view of decision making and showing how Shackle’s model of choice under uncertainty provided an alternative way of dealing with some of the issues that Kahneman and Tversky sought to address, three decades after Shackle worked out his model, via their Prospect Theory. Aside from not including ‘loss aversion’, it was Shackle’s model that was the more original and philosophically well-founded. Keywords: Prospect Theory, satisficing, bounded rationality, choice under uncertainty JEL Classification Codes: B31, D03, D81 1. Introduction In his highly successful 2011 book Thinking, Fast and Slow Daniel Kahneman offers an excellent account of his career-long research on judgment and decision- making, much of it conducted with the late Amos Tversky. Kahneman was awarded the 2002 Bank of Sweden PriZe in Economic Sciences in Memory of Alfred Nobel for this work. -
Dynamic Games Under Bounded Rationality
Munich Personal RePEc Archive Dynamic Games under Bounded Rationality Zhao, Guo Southwest University for Nationalities 8 March 2015 Online at https://mpra.ub.uni-muenchen.de/62688/ MPRA Paper No. 62688, posted 09 Mar 2015 08:52 UTC Dynamic Games under Bounded Rationality By ZHAO GUO I propose a dynamic game model that is consistent with the paradigm of bounded rationality. Its main advantages over the traditional approach based on perfect rationality are that: (1) the strategy space is a chain-complete partially ordered set; (2) the response function is certain order-preserving map on strategy space; (3) the evolution of economic system can be described by the Dynamical System defined by the response function under iteration; (4) the existence of pure-strategy Nash equilibria can be guaranteed by fixed point theorems for ordered structures, rather than topological structures. This preference-response framework liberates economics from the utility concept, and constitutes a marriage of normal-form and extensive-form games. Among the common assumptions of classical existence theorems for competitive equilibrium, one is central. That is, individuals are assumed to have perfect rationality, so as to maximize their utilities (payoffs in game theoretic usage). With perfect rationality and perfect competition, the competitive equilibrium is completely determined, and the equilibrium depends only on their goals and their environments. With perfect rationality and perfect competition, the classical economic theory turns out to be deductive theory that requires almost no contact with empirical data once its assumptions are accepted as axioms (see Simon 1959). Zhao: Southwest University for Nationalities, Chengdu 610041, China (e-mail: [email protected]). -
Arxiv:1512.06789V1 [Stat.ML] 21 Dec 2015
Information-Theoretic Bounded Rationality Pedro A. Ortega [email protected] University of Pennsylvania Philadelphia, PA 19104, USA Daniel A. Braun [email protected] Max Planck Institute for Intelligent Systems Max Planck Institute for Biological Cybernetics 72076 T¨ubingen,Germany Justin Dyer [email protected] Google Inc. Mountain View, CA 94043, USA Kee-Eung Kim [email protected] Korea Advanced Institute of Science and Technology Daejeon, Korea 305-701 Naftali Tishby [email protected] The Hebrew University Jerusalem, 91904, Israel Abstract Bounded rationality, that is, decision-making and planning under resource limitations, is widely regarded as an important open problem in artificial intelligence, reinforcement learning, computational neuroscience and economics. This paper offers a consolidated pre- sentation of a theory of bounded rationality based on information-theoretic ideas. We provide a conceptual justification for using the free energy functional as the objective func- tion for characterizing bounded-rational decisions. This functional possesses three crucial properties: it controls the size of the solution space; it has Monte Carlo planners that are exact, yet bypass the need for exhaustive search; and it captures model uncertainty arising from lack of evidence or from interacting with other agents having unknown intentions. We discuss the single-step decision-making case, and show how to extend it to sequential decisions using equivalence transformations. This extension yields a very general class of decision problems that encompass classical decision rules (e.g. Expectimax and Minimax) as limit cases, as well as trust- and risk-sensitive planning. arXiv:1512.06789v1 [stat.ML] 21 Dec 2015 c December 2015 by the authors. -
Characterizing Solution Concepts in Terms of Common Knowledge Of
Characterizing Solution Concepts in Terms of Common Knowledge of Rationality Joseph Y. Halpern∗ Computer Science Department Cornell University, U.S.A. e-mail: [email protected] Yoram Moses† Department of Electrical Engineering Technion—Israel Institute of Technology 32000 Haifa, Israel email: [email protected] March 15, 2018 Abstract Characterizations of Nash equilibrium, correlated equilibrium, and rationaliz- ability in terms of common knowledge of rationality are well known (Aumann 1987; arXiv:1605.01236v1 [cs.GT] 4 May 2016 Brandenburger and Dekel 1987). Analogous characterizations of sequential equi- librium, (trembling hand) perfect equilibrium, and quasi-perfect equilibrium in n-player games are obtained here, using results of Halpern (2009, 2013). ∗Supported in part by NSF under grants CTC-0208535, ITR-0325453, and IIS-0534064, by ONR un- der grant N00014-02-1-0455, by the DoD Multidisciplinary University Research Initiative (MURI) pro- gram administered by the ONR under grants N00014-01-1-0795 and N00014-04-1-0725, and by AFOSR under grants F49620-02-1-0101 and FA9550-05-1-0055. †The Israel Pollak academic chair at the Technion; work supported in part by Israel Science Foun- dation under grant 1520/11. 1 Introduction Arguably, the major goal of epistemic game theory is to characterize solution concepts epistemically. Characterizations of the solution concepts that are most commonly used in strategic-form games, namely, Nash equilibrium, correlated equilibrium, and rational- izability, in terms of common knowledge of rationality are well known (Aumann 1987; Brandenburger and Dekel 1987). We show how to get analogous characterizations of sequential equilibrium (Kreps and Wilson 1982), (trembling hand) perfect equilibrium (Selten 1975), and quasi-perfect equilibrium (van Damme 1984) for arbitrary n-player games, using results of Halpern (2009, 2013). -
Imbalanced Collusive Security Games
Imbalanced Collusive Security Games Han-Ching Ou, Milind Tambe, Bistra Dilkina, and Phebe Vayanos Computer Science Department, University of Southern California {hanchino,tambe,dilkina,phebe.vayanos }@usc.edu Abstract. Colluding adversaries is a crucial challenge for defenders in many real-world applications. Previous literature has provided Collusive Security Games (COSG) to model colluding adversaries, and provided models and algorithms to generate defender strategies to counter colluding adversaries, often by devising strategies that inhibit collusion [6]. Unfortunately, this previous work focused exclusively on situations with perfectly matched adversaries, i.e., where their rewards were symmetrically distributed. In the real world, however, defenders often face adversaries where their rewards are asymmetrically distributed. Such inherent asymmetry raises a question as to whether human adversaries would at- tempt to collude in such situations, and whether defender strategies to counter such collusion should focus on inhibiting collusion. To address these open ques- tions, this paper: (i) explores and theoretically analyzes Imbalanced Collusive Security Games (ICOSG) where defenders face adversaries with asymmetrically distributed rewards; (ii) conducts extensive experiments of three different adver- sary models involving 1800 real human subjects and (iii) derives novel analysis of the reason behind why bounded rational attackers models outperform perfectly rational attackers models. The key principle discovered as the result of our -
Correlated Equilibria 1 the Chicken-Dare Game 2 Correlated
MS&E 334: Computation of Equilibria Lecture 6 - 05/12/2009 Correlated Equilibria Lecturer: Amin Saberi Scribe: Alex Shkolnik 1 The Chicken-Dare Game The chicken-dare game can be throught of as two drivers racing towards an intersection. A player can chose to dare (d) and pass through the intersection or chicken out (c) and stop. The game results in a draw when both players chicken out and the worst possible outcome if they both dare. A player wins when he dares while the other chickens out. The game has one possible payoff matrix given by d c d 0; 0 4; 1 c 1; 4 3; 3 with two pure strategy Nash equilibria (d; c) and (c; d) and one mixed equilibrium where each player mixes the pure strategies with probability 1=2 each. Now suppose that prior to playing the game the players performed the following experiment. The players draw a ball labeled with a strategy, either (c) or (d) from a bag containing three balls labelled c; c; d. The players then agree to follow the strategy suggested by the ball. It can be verified that there is no incentive to deviate from such an agreement since the suggested strategy is best in expectation. This experiment is equivalent to having the following strategy profile chosen for the players by some third party, a correlation device. d c d 0 1=3 c 1=3 1=3 This matrix above is not of rank one and so is not a Nash profile. And, the social welfare in this scenario is 16=3 which is greater than that of any Nash equilibrium. -
Noisy Directional Learning and the Logit Equilibrium*
CSE: AR SJOE 011 Scand. J. of Economics 106(*), *–*, 2004 DOI: 10.1111/j.1467-9442.2004.000376.x Noisy Directional Learning and the Logit Equilibrium* Simon P. Anderson University of Virginia, Charlottesville, VA 22903-3328, USA [email protected] Jacob K. Goeree California Institute of Technology, Pasadena, CA 91125, USA [email protected] Charles A. Holt University of Virginia, Charlottesville, VA 22903-3328, USA [email protected] PROOF Abstract We specify a dynamic model in which agents adjust their decisions toward higher payoffs, subject to normal error. This process generates a probability distribution of players’ decisions that evolves over time according to the Fokker–Planck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This ‘‘logit equilibrium’’ forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data. ECTED Keywords: Bounded rationality; noisy directional learning; Fokker–Planck equation; potential games; logit equilibrium JEL classification: C62; C73 I. Introduction Small errors and shocks may have offsetting effects in some economic con- texts, in which case there is not much to be gained from an explicit analysis of stochasticNCORR elements. In other contexts, a small amount of randomness can * We gratefully acknowledge financial support from the National Science Foundation (SBR- 9818683U and SBR-0094800), the Alfred P. Sloan Foundation and the Dutch National Science Foundation (NWO-VICI 453.03.606). -
Relation Exchange As the Model of Bounded Rationality
Relation exchange as the model of bounded rationality June 16, 2017 Sung Sup Rhee Emeritus professor at Soongsil University [email protected] This paper is a preliminary version which is prepared for the presentation at 2017 KEA- APEA conference, July 7-8, 2017. Title: Relation exchange as the model of bounded rationality Author: Sung Sup Rhee/Emeritus professor at Soongsil University/[email protected] Abstracts: Kahneman (2003) presented the scheme of human cognitive system, which presents the steps of sensory order. Human cognition begins with perceptions, from which intuition comes off as fast, parallel, automatic, effortless and emotional system. Reasoning is slow, serial, controlled, effortful and rule-governed system of cognition. The architecture of rational agent model is built on the unrealistic assumption of single cognitive system which has the logical ability of a flawless system of reasoning. However, upon encountering the need to make decision, individuals are likely to act intuitively because intuition is more accessible than reason, which is at odds with the premise of rational agent model and prompts the launching of empiricism approach (Hume 1739). The process of sympathy and consent may be considered as the conduit between different cognitive systems of different individuals. Then, the relational interactions among individuals, i.e. relation exchange should be conceived as the outcome of sympathy-consent process (Rhee 2012b, 2016b). The fundamentality of relation exchange in comparison with value exchange vindicates the legitimacy of relation exchange as the model of bounded rationality. Keywords Cognitive system, accessibility, sympathy-consent process, relation exchange, open/indeterminate system, bounded rationality JEL code: D01, D03 I.